Making an impression means to leave people with a memory and opinion of who you are. You might make a good or bad impression, but they will not forget you. This is similar to 'impress', except that when you say that somebody impressed you, or was impressive, that means they made a good impression. The phrase is used intransitively. Giving an impression is similar, but used transitively to describe the leaving of a memory and opinion about something specific. For example: my face gives an impression of what I am thinking.

When you make an impression on someone, they have taken notice of you and will likely recall you later. You can make a good impression, or a bad impression on someone. To give someone an impression is to do or say something that leads them to a certain belief or point of view. This belief maybe correct, or incorrect. For example: His lack of response gave her the impression that he didn't care. So she formed a view based on his behaviour - he gave her the impression of someone who didn't care.

Jimmy explains it well. Read his first. I'd add that "give an impression" or "give the impression" are sometimes quite distinct from "make an impression'. His talk gave me the impression that Paris is best to visit in the Spring. Nothing to do with the speaker making an impression on me, which would mean that I remember him/her. . His talk left us all with an impression that oposing the plan was fruitless and would be a waste of time. The speaker's delivery however was bland and lifeless. The man himself made no substantial impression on anyone. Hope it helps.

They basically mean the same thing. I cannot think of a situation where they wouldn’t.